OK, I don't know how "common" broken engine or transmission mounts are in our trucks, but I've driven and worked on lots of vehicles with broken mounts and I know what it's like to drive them.
The engine and transmission will kick around, but I wouldn't describe it as "If I press the gas to go quickly it high revs then eventually will go into gear".
If the mounts were bad, and you hit the gas quickly, you would hear a thump as the engine lifted and hit something and a slight lag before the truck moved out. If you heal the gas down it would shift at the appropriate time, without much rpm rise between gears.
Now, having said that, I'm not saying that your mounts are NOT bad, just saying I don't think, even if they are bad, that replacing them will fix what you are experiencing. I wouldn't want you to think that simply replacing the mounts is your fix. By the sounds of it you have a transmission problem.
If your fluid level is good, then you need to find a transmission shop with a good reputation. Find one that's been in business for years. Check the BBB for problems with the shop. Get a few opinions before you commit. A knowledgeable shop will have a good idea of what is happening and what needs to be done. It could be something as simple as having a bad solenoid replaced, but it could also mean that the transmission has to come out and come apart.
I was planning on procrastinating today but I think I'll do that tomorrow.